Algebra Examples

$\frac{6}{6 + \sqrt{12 x}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $12 x$ as $2^{2} \cdot (3 x)$ .

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Step 1.1.1

Factor $4$ out of $12$ .

$\frac{6}{6 + \sqrt{4 (3) x}}$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{6}{6 + \sqrt{2^{2} \cdot 3 x}}$

Step 1.1.3

Add parentheses.

$\frac{6}{6 + \sqrt{2^{2} \cdot (3 x)}}$

Step 1.2

Pull terms out from under the radical.

$\frac{6}{6 + 2 \sqrt{3 x}}$

Step 1.3

Factor $2$ out of $6 + 2 \sqrt{3 x}$ .

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Step 1.3.1

Factor $2$ out of $6$ .

$\frac{6}{2 \cdot 3 + 2 \sqrt{3 x}}$

Step 1.3.2

Factor $2$ out of $2 \cdot 3 + 2 \sqrt{3 x}$ .

$\frac{6}{2 (3 + \sqrt{3 x})}$

Step 2

Cancel the common factor of $6$ and $2$ .

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Step 2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 3}{2 (3 + \sqrt{3 x})}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Cancel the common factor.

$\frac{2 \cdot 3}{2 (3 + \sqrt{3 x})}$

Step 2.2.2

Rewrite the expression.

$\frac{3}{3 + \sqrt{3 x}}$

Step 3

Multiply $\frac{3}{3 + \sqrt{3 x}}$ by $\frac{3 - \sqrt{3 x}}{3 - \sqrt{3 x}}$ .

$\frac{3}{3 + \sqrt{3 x}} \cdot \frac{3 - \sqrt{3 x}}{3 - \sqrt{3 x}}$

Step 4

Multiply $\frac{3}{3 + \sqrt{3 x}}$ by $\frac{3 - \sqrt{3 x}}{3 - \sqrt{3 x}}$ .

$\frac{3 (3 - \sqrt{3 x})}{(3 + \sqrt{3 x}) (3 - \sqrt{3 x})}$

Step 5

Expand the denominator using the FOIL method.

$\frac{3 (3 - \sqrt{3 x})}{9 - 3 \sqrt{3 x} + \sqrt{3 x} \cdot 3 - {\sqrt{3 x}}^{2}}$

Step 6

Simplify.

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Step 6.1

Rewrite ${\sqrt{3 x}}^{2}$ as $3 x$ .

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Step 6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

$\frac{3 (3 - \sqrt{3 x})}{9 - {({(3 x)}^{\frac{1}{2}})}^{2}}$

Step 6.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 (3 - \sqrt{3 x})}{9 - {(3 x)}^{\frac{1}{2} \cdot 2}}$

Step 6.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 (3 - \sqrt{3 x})}{9 - {(3 x)}^{\frac{2}{2}}}$

Step 6.1.4

Cancel the common factor of $2$ .

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Step 6.1.4.1

Cancel the common factor.

$\frac{3 (3 - \sqrt{3 x})}{9 - {(3 x)}^{\frac{2}{2}}}$

Step 6.1.4.2

Rewrite the expression.

$\frac{3 (3 - \sqrt{3 x})}{9 - {(3 x)}^{1}}$

Step 6.1.5

Simplify.

$\frac{3 (3 - \sqrt{3 x})}{9 - (3 x)}$

Step 6.2

Multiply $3$ by $- 1$ .

$\frac{3 (3 - \sqrt{3 x})}{9 - 3 x}$

Step 7

Cancel the common factors.

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Step 7.1

Factor $3$ out of $9$ .

$\frac{3 (3 - \sqrt{3 x})}{3 (3) - 3 x}$

Step 7.2

Factor $3$ out of $- 3 x$ .

$\frac{3 (3 - \sqrt{3 x})}{3 (3) + 3 (- x)}$

Step 7.3

Factor $3$ out of $3 (3) + 3 (- x)$ .

$\frac{3 (3 - \sqrt{3 x})}{3 (3 - x)}$

Step 7.4

Cancel the common factor.

$\frac{3 (3 - \sqrt{3 x})}{3 (3 - x)}$

Step 7.5

Rewrite the expression.

$\frac{3 - \sqrt{3 x}}{3 - x}$